Critical gravitational collapse of a perfect fluid: Nonspherical perturbations

Continuously self-similar (CSS) solutions for the gravitational collapse of a spherically symmetric perfect fluid, with the equation of state p=κρ, with 0<κ<1 a constant, are constructed numerically and their linear perturbations, both spherical and nonspherical, are investigated. The l=1 axial perturbations admit an analytical treatment. All others are studied numerically. For intermediate equations of state, with 1/9<κ≲0.49, the CSS solution has one spherical growing mode, but no nonspherical growing modes. That suggests that it is a critical solution even in (slightly) nonspherical collapse. For this range of κ we predict the critical exponent for the black hole angular momentum to be 5(1+3κ)/3(1+κ) times the critical exponent for the black hole mass. For κ=1/3 this gives an angular momentum critical exponent of μ≃0.898, correcting a previous result. For stiff equations of state, 0.49≲κ<1, the CSS solution has one spherical and several nonspherical growing modes. For soft equations of state, 0<κ<1/9, the CSS solution has 1+3 growing modes: a spherical one, and an l=1 axial mode (with m=-1,0,1).